2014 PJC P2 Q9

A female doctor wishes to test a new treatment for a certain genetic disease. On a particular day, she asks \(n\) (where \(n\) is fixed) patients who come to her clinic suffering from this disease if they are willing to take part in a trial. Based on past experiences, she finds that the probability of such a patient willing to take part is \(0.3\). The number of patients who are willing to take part is the random variable \(R\).

1. State, in the context of this question, two assumptions needed to model \(R\) by a binomial distribution.[2]
2. Explain why one of the assumptions stated in part (i) may not hold in this context.[1]

Assume now that these assumptions do in fact hold.

3. Given that \(n=8\), find the probability that \(R\) is at least \(2\).[1]
4. If the chance of at least \(2\) patients willing to take part in this trial is more than \(90\%\), find the least possible integer value of \(n\).[3]